Operator Definition and 1000 Threads

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).
This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.

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  1. L

    MHB Linear operator, its dual, proving surjectivity

    Let T: X \rightarrow Y be a continuous linear operator between Banach spaces. Prove that $T$ is surjective \iff T^* is injective and im T^* is closed. I've proven a "similar" statement, with imT^* replaced with imT. There I used these facts: $\overline{imT}= ^{\perp}(kerT^*)$ and...
  2. P

    Question about the Hamiltonian Operator

    In my physical chemistry course, we are learning about the Schrödinger Equation and were introduced to the Hamiltonian Operator recently. We started out with the simple scenario of a particle in 1D space. Our professor's slide showed the following "derivation" to arrive at the expression for the...
  3. Medicol

    How Can I Compute Variable B from Variable A Using Inverse OR Operator?

    I have 2 variables A and B. A can be computed from B like this Input: BOOL b, OBJECT B A=(B<<5) if NOT b then A=[A OR 0xFF] Now I would like to compute B from the above code Input: BOOL b, OBJECT A It can be either B=A>>5 if NOT b then B= ? or if NOT b then B=? B=A>>5 I am not...
  4. K

    Operator state mapping in Conformal Field Theory

    can anyone suggest any good reading material on operator state mapping in conformal field theory? I know only elementary field theory... So it might be helpful ifsomeone suggest a book where it is done in little detailed way..
  5. H

    How to treat spin orbit operator directly

    For an electron the spin operator S_zis represented by a 2×2 matrix, with spin up and down as its bases. Consider the angular momentum operator L_z with l=1 which is a 3×3 matrix. How can we treat the L_z S_z operator directly in matrix form?
  6. P

    Time-energy uncertainty and derivative of an operator

    Homework Statement I would appreciate feedback on the following two problems: (1) For a given operator A with no explicit time dependence I am asked to show that d/dt(eAt)=A(eAt) (2) A free wave packet of width Δx is traveling at a constant velocity v0=p0/m. I am asked to estimate the...
  7. D

    Irreducible linear operator is cyclic

    I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...
  8. nomadreid

    Hermitian operator represented as a unitary operator

    Homework Statement I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary...
  9. blue_leaf77

    Momentum operator eigenfunction

    This might be trivial for some people but this has been bothering lately. If P is momentum operator and p its eigenvalue then the eigenfunction is up(x) = exp(ipx/h). where h is the reduced Planck constant (sorry can't find a way to make the proper notation). While it can also be proved that...
  10. P

    Matrix representation of an operator in a new basis

    Homework Statement Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. I am asked to write down the matrix representation of A in the new basis. Homework EquationsThe Attempt at a Solution...
  11. Fantini

    MHB Momentum operator in the position representation

    Hi! :) I'm trying to understand the following calculation. The book Quantum Mechanics by Nouredine Zettili wants to determine the form of the momentum operator $\widehat{\vec{P}}$ in the position representation. To do so he calculates as follows: $$\begin{aligned} \langle \vec{r} |...
  12. W

    Linear operator and linear vector space?

    hi, please tell me what do we mean when we say in quantum mechanics operators are linear and also vector space is also linear ?
  13. binbagsss

    Kdv solution solitons Bilinear Operator

    Homework Statement [/B] I'm solving for ##f_1## from ##B(f_{1}.1+1.f_{1})## from ## \frac{\partial}{\partial x}(\frac{\partial}{\partial t}+\frac{\partial^{3}}{\partial x^{3}})f_n=-\frac{1}{2}\sum^{n-1}_{m=1}B(f_{n-m}.f_{m}) ## where ##B=D_tD_x+D_x^4##, where ##B## is the Bilinear...
  14. A

    Time ordering operator, interaction Lagrangian, QED

    Homework Statement I am trying to calculate the following quantity: $$<0|T\{\phi^\dagger(x_1) \phi(x_2) exp[i\int{L_1(x)dx}]\}|0>$$ where: $$ L_1(x) = -ieA_{\mu}[\phi^* (\partial_\mu \phi ) - (\partial_\mu \phi^*)\phi] $$[/B] I am trying to find an expression including the propagators...
  15. M

    Measuring quark colour operator

    What is the explicit 3x3 matrix operator which measures the colour of a quark? Essentially what I want to know is what is the analogue of ##S^z## for the measuring of spin.
  16. G

    Linear operator, linear functional difference?

    What is a difference between linear operator and linear functional? Do I understand it correctly that linear operator is any operator that when applied on a vector from a vector space, gives again a vector from this vector space. And also obeys linearity conditions. And linear functional is a...
  17. M

    Checking if Momentum Operator is Hermitian - Integration

    Homework Statement I'm checking to see if the momentum operator is Hermitian. Griffiths has the solution worked out, I'm just not following the integration by parts. Homework Equations int(u dv) = uv - int(v du) The Attempt at a Solution I've attached an image of my work. It seems there...
  18. I

    C/C++ C++ *Pointer vs. Pointer* and Member Access Operator

    A pointer in C++ is represented by *. Sometimes the * comes after the variable/class/whatever such as 'Pointer*'. Other times it comes before, '*Pointer'. What is the difference between the two?What is the member access operator for? (->) According to my notes, a->b is equivalent to (*a).b
  19. M

    Temporal component of the normal ordered momentum operator

    Homework Statement Consider the real scalar field with the Lagrangian \mathcal{L}=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. Show that after normal ordering the conserved four-momentum P^\mu = \int d^3x T^{0 \mu} takes the operator form P^\mu = \int...
  20. T

    Angular Momentum squared operator (L^2) eigenvalues?

    Homework Statement Find the eigenvalues of the angular-momentum-squared operator (L2) for hydrogen 2s and 2px orbitals... Homework Equations Ψ2s = A (2-r/a0)e-r/(2a0) Ψ2px = B (r/a0)e-r/(2a0) The Attempt at a Solution If I am not wrong, is the use of L2 in eigenfunction L2Ψ = ħ2 l(l+1) Ψ...
  21. DavideGenoa

    Eigenvectors of Fourier transform operator #F:L^2\to L^2#

    Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...
  22. ShayanJ

    Momentum operator in "open" space

    Consider the Hilbert space H=L^2([0,1],dx) . Now we define the operator P=\frac \hbar i \frac{d}{dx} on this Hilbert space with the domain of definition D(P)=\{ \psi \in H | \psi' \in H \ and \ \psi(0)=0=\psi(1) \} . Then it can be shown that P^\dagger=\frac \hbar i \frac{d}{dx} with...
  23. B

    Showing an operator is well-defined

    Homework Statement Let ##H## be a normal subgroup of a group ##(G, \star)##, and define ##G/H## as that set which contains all of the left cosets of ##H## in ##G##. Define the binary operator ##\hat{\star}## acting on the elements of ##G/H## as ##g H \hat{\star} g' K = (g \star g') H##...
  24. ShayanJ

    Confusion about position operator in QM

    In quantum mechanics, the position operator(for a single particle moving in one dimension) is defined as Q(\psi)(x)=x\psi(x) , with the domain D(Q)=\{\psi \epsilon L^2(\mathbb R) | Q\psi\epsilon L^2 (\mathbb R) \} . But this means no square-integrable function in the domain becomes...
  25. D

    Question about d'Alembert operator

    Hey guys, The expression \partial_{\mu}\partial^{\nu}\phi is equal to \Box \phi when \mu = \nu. However when they are not equal, is this operator 0? I'm just curious cos this sort of thing has turned up in a calculation of mine...if its 0 I'd be a very happy boy
  26. G

    Quantizing the conjugate operator to adjoint operator

    If you have the product of two Grassman numbers C=AB, and take the conjugate, should it be C*=A*B*, or C*=B*A*? The general rule for operators, whether they are Grassman operators (like the Fermion field operator) or the Bose field operator, I think is (AB)^dagger=B^dagger A^dagger. This...
  27. D

    Derivative of d'Alambert operator?

    Hi guys, So I've ended up in a situation where I have \partial_{\mu}\Box\phi. where the box is defined as \partial^{\mu}\partial_{\mu}. I'm just wondering, is this 0 by any chance...? Thanks!
  28. BiGyElLoWhAt

    The Differential operator and it's limitations

    So I had a quiz today, and one of the questions was pretty easy, pretty straight forward. Show that ##t^2e^{9t}## is a solution of ##(D-9)^3y## Foil it out, plug in y, and you're done. Well I tried doing something else, that (at least in my mind) should have worked, but it didn't. I said...
  29. A

    What is the suitable unitary operator for a rotating frame?

    Hello, I have a Hamiltonian that describes a particle in a rotating cylindrical container at angular frequency ω. In the lab frame the Hamiltonian is time-dependent and takes the form (using cylindrical coordinates) \mathcal H_o=\frac{\vec P^2}{2m}+V(r,\theta-\omega t,z), where V(r,\theta,z)...
  30. D

    Klein-Gordon operator on a time-ordered product

    Homework Statement Hey guys, So here's the problem I'm faced with. I have to show that (\Box + m^{2})<|T(\phi(x)\phi^{\dagger}(y))|>=-i\delta^{(4)}(x-y) , by acting with the quabla (\Box) operator on the following...
  31. D

    D'Alambertian operator on the heaviside function?

    Hey guys, How does one compute the following quantity: \Box \theta(x_{0})=\partial_{0}\partial^{0}\theta(x_{0})? I know that \partial_{0}\theta(x_{0})=\delta(x_{0}) which is the Dirac delta, but what about the second derivative? Thanks everyone!
  32. Breo

    About Wick's Theorem, Time Order Operator, Normal Ordering and Green's Function

    So if I understood well, Normal ordering just comes due to the conmutation relation of a and a⁺? right? Is just a simple and clever simplification. Wick Theorem is analogue to normal ordering because it is related to the a and a⁺ again (so related to normal ordering, indeed). However I do not...
  33. S

    How Do Fourier Integral Operators Work in Mathematical Analysis?

    Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator: $$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ where $$Au\in C^0...
  34. L

    Does this curl operator equal 1?

    Hello, I am a beginner in electromagnetism. I am trying to find a vector field whose rotation equals 1 with a curl operator. If I say that the vector field is defined by V(y;2x;0) does it work? As a result, I find (0;0;1), am I right?
  35. D

    AC circuits -- Why we introduce the J operator in analyzing them

    I am just wondering why or how we introduce the J operator in analyzing ac circuits. I want more of a proof for this.
  36. perplexabot

    How Do You Change the Basis of an Operator in Linear Algebra?

    Homework Statement Homework Equations \check{T} = BTB^{-1} (eq1) The Attempt at a Solution Ok, so I have a couple of questions here if I may ask... First, I want to be sure I understand the wording of (a) and (b) correctly. Is the following true?: (a) Write the matrix T...
  37. R

    Modulus & Division: Last Digit of Numbers Explained

    Isn't it amusing ?What could be the probable explanation for this?Also when operated by division operator gives the rest of the number as the quotient (Note only when the divisor is 10)
  38. Xemnas92

    Grand Canonical Ensemble: N operator problem

    I have a problem in understanding the quantum operators in grand canonical ensemble. The grand partition function is the trace of the operator: e^{\beta(\mu N-H)} (N is the operator Number of particle) and the trace is taken on the extended phase space: \Gamma_{es}= \Gamma_1 \times \Gamma_2...
  39. maverick280857

    Conformal weights of the vertex operator

    Hi, I'm trying to prove that the conformal weight of the bosonic vertex operator :e^{ik\cdot X}: is \left(\frac{\alpha'k^2}{4},\frac{\alpha'k^2}{4}\right). I've done some algebra but I think I am making some mistake with a factor of 2 somewhere because I get a 1/2 instead of a 1/4. My attempt...
  40. C

    The simplest derivation of position operator for momentum space

    Might be simple but I couldn't see. We can easily derive momentum operator for position space by differentiating the plane wave solution. Analogously I want to derive the position operator for momentum space, however I am getting additional minus sign. By replacing $$k=\frac{p}{\hbar}$$ and...
  41. KleZMeR

    How to Simplify the Pauli Matrix Expression with the Del Operator?

    Homework Statement In the Pauli theory of the electron, one encounters the expresion: (p - eA)X(p - eA)ψ where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ...
  42. baby_1

    Simple question in Del operator on plane wave equation

    Hello question is: As you see when we do del operator on A vector filed in below example it removes exponential form at the end.why does it remove exponential form finally?
  43. kini.Amith

    Identifying Projection Operators: Is Idempotence Enough?

    If we are given an operator, say in matrix or outer product form, then how can we check if it is a projection operator? Is idempotence a sufficient condition for an operator to be a projection operator or are there any other conditions?
  44. T

    Finding an Operator (from a textbook)

    Homework Statement Because I wanted to practice more of operators, I borrowed a textbook from a library for extra problems...I managed to solve (a) to (e), but not the last question...which is: Write out the operator A2 for A: (f) d2/dx2 - 2xd/dx + 1 for which I keep getting a different...
  45. T

    How to Find the Operator A2 for A = d/dx + x

    Hello--I am practicing for the upcoming quiz (as part of quantum physics), but have no idea how to solve the sample problem the teacher gave (w/ solution)... Write out the operator A2 for A = d/dx + x (Hint: INCLUDE f(x) before carrying out the operations) so...I tried: A {df(x)/dx + f(x)} =...
  46. arcoon

    Is (i/x^2 d/dx) a Hermitian Operator?

    Homework Statement Hi, I'm doing a Quantum chemistry and one of my question is to determine if is hermitian or not? I am learning and new to this subject... Cant figure out how to do this question at all. Please helppp! ^Q= i/x^2 d/dx is hermitian or not? Homework Equations The Attempt at a...
  47. F

    MHB What is "identical to" operator?

    What does mean "identically equal to" in the context of differential equations? In class the prof wrote \mu_x \equiv 0. I asked what it meant and he said "it means identical to". Can someone elaborate, for example what purpose does it surve? If it just means a function always has that value, why...
  48. O

    Definition Of Operator Exponential?

    I've been reading Mermin's book on Quantum Computer Science, and in the section in which he discusses the construction of a QFT using 1-Qbit and 2-Qbit gates, he makes reference to some expressions involving linear operators that I'm not familiar with (at least if I've seen them before I've...
  49. A

    Eigenvalues and Eigenvectors of a Hermitian operator

    Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...
  50. M

    Commutator of Boost Generator with Creation operator

    Homework Statement Given that U upon acting on the creation operator gives a creation operator for the transformed momentum $$U(\Lambda) a_p^\dagger U(\Lambda)^\dagger = a_{\boldsymbol{\Lambda} \mathbf{p}}^\dagger $$ and ##\Lambda ## is a pure boost, that is ## U(\Lambda) = e^{i...
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